Gapless mean-field theory of Bose-Einstein condensates

We present a topical review of the development of finite-temperature field theories of Bose-Einstein condensation in weakly interacting atomic gases. We highlight the difficulties in obtaining a consistent finite-temperature theory that has a gapless excitation spectrum in accordance with Goldstone' s theorem and which is free from both ultraviolet and infrared divergences. We present results from the two consistent theories developed so far. Thes e are the Hartree-Fock-Bogoliubov theory within the Popov approximation and a many-body T-matrix approach which we have termed gapless-Hartree-Fock-Bo goliubov (GHFB). Comparison with the available experimental results is made and the remaining difficulties are highlighted.